Problem: For what value of $x$ will $\frac{3+x}{5+x}$ and $\frac{1+x}{2+x}$ be equal?
Solution: We have the equation $\frac{3+x}{5+x}=\frac{1+x}{2+x}$. Simplifying, we get \begin{align*}
(3+x)(2+x)&=(5+x)(1+x)\\
6+5x+x^2&=5+6x+x^2\\
x&=\boxed{1}.
\end{align*}